Electron density studies on magnetic systems

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Delft University of Technology

Electron density studies on magnetic systems

Boeije, Maurits DOI 10.4233/uuid:f0f06ff5-53b9-4a79-87c9-f49aee5fd405 Publication date 2017 Document Version Final published version Citation (APA)

Boeije, M. (2017). Electron density studies on magnetic systems. https://doi.org/10.4233/uuid:f0f06ff5-53b9-4a79-87c9-f49aee5fd405

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ELECTRON DENSITY STUDIES

ON MAGNETIC SYSTEMS

MAURITS BOEIJE

ELEC

TRON DENSIT

Y S

TUDIES ON MA

GNETIC S

YS

TEMS

MA

URIT

S BOEIJE

9 789462 956353

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Electron density studies on magnetic systems

Proefschrift

ter verkrĳging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magniﬁcus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 3 juli 2017 om 12:30 uur

door

Maurits Boeije

Master of Science in Chemistry, Radboud Universiteit, Nĳmegen geboren te Oostburg, Nederland.

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Dit proefschrift is goedgekeurd door de promotor: prof. dr. E.H. Brück copromotor: dr. ir. N.H. van Dĳk Samenstelling promotiecommissie:

Rector Magniﬁcus, voorzitter

Prof. dr. E.H. Brück, Technische Universiteit Delft Dr. ir. N.H. van Dĳk, Technische Universiteit Delft Onafhankelĳke leden:

Prof. dr. L.F. Cohen, Imperial College London

Dr. ir. G.A. de Wĳs, Radboud Universiteit Nĳmegen

Prof. dr. P. Dorenbos, Technische Universiteit Delft Prof. dr. F.M. Mulder, Technische Universiteit Delft Prof. dr. ir. T.H. van der Meer, Universiteit Twente

The work presented in this PhD thesis is ﬁnancially supported by the Foundation of Fundamental Research on Matter (FOM) via the Industrial Partnership Program IPP I28 and co-ﬁnanced by BASF New Business.

Printed by: BOXPress

Front: Artist impression of the magnetoelastic transition in Fe2P-based

mate-rials. Electrons are shown (red, blue and purple) on a hexagonal lattice of atoms (green and orange) that are represented by their core electron cloud. On the bottom of the image, the material is metallic; the elec-trons are aligned ferromagnetically and are present in bands (blue). Above the Curie temperature, on the top of the image, the electrons become localized and pair up, reducing the magnetic moment and in-creasing the bond strength.

An electronic version of this dissertation is available at http://repository.tudelft.nl/.

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Het leven draait niet om het halen van doelen. Het leven draait om hard werken en doen waar je goed in bent. Dan zal je op een dag terugkĳken op je prestaties en gelukkig zĳn.

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1

Introduction

1.1. Motivation

R

efrigeration technology has had a great impact on everyday life since it became widely available in the twentieth century. It increased the living standard by allow-ing food to be stored for a longer time and decreased illness due to food spoilage1_{. Apart} from these positive effects, refrigeration based on vapor compression posed new risks to human health due to the used refrigeration gases. The used gases are generally toxic, detrimental for the ozone layer or are greenhouse gases2_{. This development sparked an} interest in refrigeration techniques that do not use the gas-to-liquid phase transition, but focus on the solid-to-solid phase transition. One of the phenomena that can be used in solids is the magnetocaloric effect. Efficiency studies show that a cooling cycle using magnetocaloric materials is also more efficient compared to vapor-compression tech-niques3_{. With a growing global demand for refrigeration}4,5_{, the increased efficiency} can further decrease the rate by which greenhouse gases are emitted. To maximize the magnetocaloric effect, one has to find and optimize new material systems.

1.2. Magnetocaloric eﬀect

M

agnetic refrigeration is based on the magnetocaloric effect (MCE), that is charac-terized by the isothermal entropy change and adiabatic temperature change re-sulting from a change in applied magnetic ﬁeld. The entropy change can be realized in ferromagnetic6 _{or paramagnetic}7 _{materials and is most pronounced at the magnetic} transition temperature. In general, the transition is of second order showing a discon-tinuous change of the heat capacity due to the magnetic transition.

By expressing the entropy 𝑆 in terms of the heat capacity 𝐶 𝑆 = ∫

𝑇

0

𝐶

𝑇𝑑𝑇 , (1.1)

one can construct a schematic drawing that clearly illustrates the magnetocaloric effect for materials having a second-order phase transition, shown inFigure 1.1. When the

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1

2 1.Introduction

material is in the paramagnetic state, the application of an external magnetic ﬁeld shifts the critical point and the material can become ferromagnetic, lowering the heat capacity.

Figure 1.1: Schematic diagram of a second-order phase transition where there is an adiabatic temperature change ∆𝑇𝑎𝑑due to a change in heat capacity ∆𝐶. In the paramagnetic state, the heat capacity is large and

the entropy 𝑆 is given by the surface of the black box. Due to the application of a magnetic ﬁeld, the material becomes ferromagnetic, with a lower associated heat capacity. The total entropy can initially be kept constant, in accordance with an adiabatic process. The resulting temperature change can subsequently interact with the environment, lowering the entropy of the material.

There are three main contributions to the entropy and heat capacity: structural, mag-netic and electronic. A change in magnetization gives rise to a change in heat capacity and thus gives rise to the MCE. This is most pronounced for rare earth-based materials that have a high magnetization associated with the localized 4𝑓 orbitals. For example, ErAl2shows a Δ𝑇𝑎𝑑of 6 K for a ﬁeld change of 0 − 2 T8. The low transition

temper-atures of rare earth-based materials are usually between 5 and 100 K and make these materials unsuitable for room-temperature applications. The only candidate is Gd with a transition temperature of 292 K, but the cost and limited availability of this material make it impractical to use in large-scale applications.

Instead, cheap and abundant 3𝑑-based intermetallic compounds with a high mag-netization and a 𝑇𝐶 around room temperature are preferred for room temperature

ap-plications. Because the structural heat capacity saturates to the Dulong-Petit limit (𝐶 = 3R) at the Debye temperature (generally between 400 and 500 K), the MCE for these materials is generally small. Instead of using materials showing a second-order phase transition, materials with a ﬁrst-order phase transition (giant MCE) are used. Especially Fe2P-based materials are interesting because of their high magnetization, high entropy

change and magnetoelastic transition.

For these compounds, there is a coupled magnetic and crystallographic transition, without changing the crystal symmetry. In addition to a Δ𝐶 of about 1%9_{, there is a} simultaneous 𝑒𝑛𝑡ℎ𝑎𝑙𝑝𝑦 change at the transition, or latent heat, given by Δ𝐻 = 𝑇 Δ𝑆. This can triple the Δ𝑇𝑎𝑑to about 4.2 K, measured in a ﬁeld change from 0 − 1.5 T

for FeMn(P,As)10_{. For these materials, the discontinuous change in entropy associated} with ﬁrst-order phase transitions allows tapping into all three entropy reservoirs (lattice, magnetic, and electronic contributions) of the material11_.

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1.3.esis outline ..

1

3 In general, the isothermal entropy change and adiabatic temperature change are given by Δ𝑆𝑖𝑡 = ∫ 𝐵𝑓 𝐵𝑖 (𝜕𝑀 𝜕𝑇 )_𝐵𝑑𝐵, (1.2) Δ𝑇_𝑎𝑑= − ∫ 𝐵𝑓 𝐵𝑖 𝑇 𝐶𝑝 (𝜕𝑀 𝜕𝑇 ) 𝑑𝐵, (1.3)

where 𝐵𝑖 and 𝐵𝑓 are the initial and ﬁnal applied magnetic ﬁelds. Finding the

mi-croscopic origin of the coupling between magnetism and the crystal structure is not trivial and is up to now only observed in Mn(As,Sb)12_{, La(Fe,Si)}

13and its hydrides13,

(Mn,Fe)2(P,X) where X = As14, Ge15, Si16 , MnCoGe17, Gd5(Ge,Si)418, FeRh19 and

Heusler alloys20. In order to ﬁnd new promising systems, the origin of this coupling must be understood. This can yield boundary conditions that can guide the quest for new systems.

1.3. esis outline

B

y using diffraction techniques, the magnetoelastic coupling in Fe2P-based

magne-tocaloric materials as well as in a new material system has been investigated. The fundamental knowledge obtained by the derived and calculated electron density helps to establish the boundary conditions needed for ﬁnding new and effective magnetocaloric materials.

First, the role that the (symmetry of the) crystal structure plays in these compounds is discussed inchapter 2. Here the relation between the crystal structure, electronic struc-ture and magnetic properties is introduced. The experimental methods and instruments used to synthesize and characterize the structural and magnetic properties are described

inchapter 3.

The next four chapters discuss three different material classes. The ﬁrst material, dis-cussed inchapter 4, is based on Mn3Ga. For this material a simultaneous crystallographic

and magnetization change was reported. The microscopic origin is elucidated by X-ray and neutron diffraction techniques and the applicability of the MCE of this transition is discussed. The second material is discussed inchapter 5andchapter 6and is based on Fe2P. Inchapter 5, high-resolution X-ray powder diffraction was used to study the

elec-tron density in the ferro- and paramagnetic states. Experimental evidence was found to support the theory of mixed magnetism and shows the importance of the presence of different lattice sites occupied by magnetic atoms. The crystallographic properties are discussed inchapter 6and show that the magnetoelastic coupling in materials showing ﬁrst-order and second- order phase transitions is essentially the same. YFe5is discussed

inchapter 7, it is chosen for the high concentration of magnetic atoms and has two

differ-ent lattice sites for magnetic atoms. Although it is expected to crystallize in a CaCu5type

crystal structure, single phase samples could not be attained. To understand the phase stability of these compounds, an empirical model based on the Miedema parameters is introduced. This last study shows how the nature of the elements can facilitate/hinder

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1

4 References

the formation of interesting phases and can provide boundary conditions to select new promising materials.

References

[1] L. A. Craig, B. Goodwin and T. Grennes. ‘The Effect of Mechanical Refrigeration on Nutrition in the United States.’ Social Science History volume 28, pp. 325–336 (2004).

[2] A. McCulloch, P. M. Midgley and P. Ashford. ‘Releases of refrigerant gases (CFC-12, HCFC-22 and HFC-134a) to the atmosphere.’ Atmospheric Environment vol-ume 37, pp. 889 – 902 (2003).

[3] C. Zimm, A. Jastrab, A. Sternberg et al. Description and Performance of a Near-Room Temperature Magnetic Refrigerator, Springer US, pp. 1759–1766 (1998). [4] N. Wilson. ‘Magnetic Cooling: Camfridge achievements and industry challenges,

address at Thermag VII, Turin.’ (2016).

[5] D. Coulomb, J.-L. Dupont and A. Pichard. ‘The Role of Refrigeration in the Global Economy.’ IIR Notes volume 29 (2015).

[6] P. Weiss and A. Piccard. ‘Le phénomène magnétocalorique.’ J. Phys. volume 5, pp. 103–109 (1917).

[7] W. F. Giauque and D. P. MacDougall. ‘Attainment of Temperatures Below 1∘

Abso-lute by Demagnetization of Gd2(SO4)3⋅ H2O.’ Phys. Rev. volume 43, pp. 768–768

(1933).

[8] P. J. von Ranke, V. K. Pecharsky and K. A. Gschneidner Jr. ‘Inﬂuence of the crys-talline electrical ﬁeld on the magnetocaloric effect of DyAl2, ErAl2, and DyNi2.’

Phys. Rev. B volume 58, pp. 12110–12116 (1998).

[9] P. Roy, E. Brück and R. A. de Groot. ‘Latent heat of the ﬁrst-order magnetic tran-sition of MnFeSi0.33P0.66.’ Phys. Rev. B volume 93, p. 165101 (2016).

[10] E. Brück, M. Ilyn, A. M. Tishin and O. Tegus. ‘Magnetocaloric effects in MnFeP1−𝑥As𝑥-based compounds.’ Journal of Magnetism and Magnetic

Materi-als volume 290–291, Part 1, pp. 8 – 13 (2005).

[11] M. F. J. Boeĳe, P. Roy, F. Guillou et al. ‘Efﬁcient Room-Temperature Cooling with Magnets.’ Chem. Mater. volume 28, p. 4901−4905 (2016).

[12] H. Wada and Y. Tanabe. ‘Giant magnetocaloric effect of MnAs1−𝑥Sb𝑥.’ Appl.

Phys. Let. volume 79, pp. 3302–3304 (2001).

[13] A. Fujita, S. Fujieda, Y. Hasegawa and K. Fukamichi. ‘Itinerant-electron metamag-netic transition and large magnetocaloric effects in LaFe13−𝑥Si𝑥compounds and

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References ..

1

5 [14] E. Brück, M. Ilyn, A. M. Tishin and O. Tegus. ‘Magnetocaloric effects in MnFeP1−𝑥As𝑥-based compounds.’ J. Magn. Magn. Mater. volume 290-291, pp.

8–13 (2005).

[15] N. T. Trung, Z. Q. Ou, T. J. Gortenmulder et al. ‘Tunable thermal hysteresis in MnFe(P,Ge) compounds.’ Appl. Phys. Lett. volume 94, p. 102513 (2009).

[16] O. Tegus, E. Brück, K. H. J. Buschow and F. R. de Boer. ‘Transition-metal-based magnetic refrigerants for room-temperature applications.’ Nature volume 415, pp. 150–152 (2002).

[17] N. T. Trung, L. Zhang, L. Caron et al. ‘Giant magnetocaloric effects by tailoring the phase transitions.’ Appl. Phys. Lett. volume 96, p. 172504 (2010).

[18] K. A. Gschneidner Jr. and V. K. Pecharsky. ‘Giant magnetocaloric effect in Gd5Si2Ge2.’ Phys. Rev. Lett. volume 78, pp. 4494–4497 (1997).

[19] S. A. Nikitin, G. Myalikgulyev, A. M. Tishin et al. ‘The magnetocaloric effect in Fe49Rh51compound.’ Physics Letters A volume 148, pp. 363 – 366 (1990).

[20] J. Liu, T. Gottschall, K. P. Skokov et al. ‘Giant magnetocaloric effect driven by structural transitions.’ Nature Mat. volume 11, p. 620–626 (2012).

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2

eoretical background

2.1. Crystal chemistry

I

n solids, there are four main cohesive interactions: metallic, covalent, ionic and van der Waals interactions. For the three strongest, the degree to which the interaction is present in a compound can be estimated by using the concept of electronegativity (EN). This was ﬁrst introduced by Linus Pauling as being the difference in energy between the average bond strength of X-X and H-H compared to X-H, where X is the element of interest and H is used as reference1_{. These values can be found in the appendix.}

These interactions are one of the main driving forces for the arrangement of atoms on a lattice. This principle can also be reversed: the interaction between atoms can be gauged by looking at the crystal structure. Compounds with equal and high EN, attain the lowest energy by pairing electrons with neighbors, giving rise to covalent bonds. This is the case for C, where in diamond all four valence electrons are bound and all C atoms have four nearest neighbors. This gives rise to a low number of nearest neighbors and speciﬁc bond angles. In contrast, compounds with equal and low EN, attain the lowest energy when the electrons are delocalized. For fcc Fe, the electrons are shared between 12 nearest neighbors in a close-packed atomic arrangement. Compounds with more than one atomic species and different EN values ionize. This gives rise to a packing where all anions are paired with cations, according to their stoichiometry, to maximize the Coulomb interaction. The most interesting compounds, from high-𝑇𝑐

superconduc-tors like (Ba,K)Fe2As22to magnetocaloric materials like Fe2P3, show a mix of these

in-teractions, as is made clear inchapter 5. In this thesis the focus will be on intermetallics. This class of materials shows a large variety of different possible (close-packed) atomic arrangements. In order to understand this class of materials, one needs to understand and classify the different crystal structures that can be adopted.

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2

8 2.eoretical background

2.1.1. Crystal structures of intermetallic compounds

Intermetallic compounds can form in various compositional ranges. Unlike covalent and ionic solids, where the stoichiometry is ﬁxed by the number of bonds or the charge, the observed stoichiometries in intermetallic compounds is distinctly different. The complexity of intermetallic compounds can be illustrated when subjecting covalent or ionic solids to high pressure. Because this induces a close packing of the atoms, the material becomes metallic. In the case of the archetypical salt NaCl, additional phases like Na3Cl, Na2Cl, Na3Cl2 are stable with excess Na, and NaCl3, NaCl7 are stable in

excess Cl4_{. These stiochiometries, that defy chemical intuition, are very common for} intermetallic compounds and need to be described using a different classiﬁcation. This is done by using a prototype compound that describes the atomic arrangement.

For a hexagonal close-packed (hcp) crystal structure, the packing of Mg is used as prototype. Several modiﬁcations can be distinguished that lead to different prototypes. The most common modiﬁcations are shown inFigure 2.1. The type of transformations are listed inTable 2.1. For example, an ordered substitution of ¼ of the atoms in Mg will lead to the Mg3Cd prototype, discussed inchapter 4. Multiple substitutions lead to

the MgZn2Laves phase, while a redistribution leads to the CaCu5prototype, discussed

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2.1.Crystal chemistry ..

2

9

Figure 2.1: Relationships between structure types of intermetallic compounds5. The color code corresponds to the types of transformation indicated with roman numerals.

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2

10 2. eor etical backgr ound

Table 2.1: Relationships between structure types of intermetallic compounds showing common features and ways of transformation5_.

Common features of structure Ways of transforming Examples of structure types

types Parent → transformed structure

Equal positions for all atoms Ia Ordered substitution Cu → AuCu3, ThMn12→ CeMn4Al8

(superstructure formation)

Ib Redistribution of the different MgCu2→ AuBe5, MgSnCu4

component atoms

Approximately equal positions II External deformationb _{Cu → 𝛼-Mn}

for all atoms Internal deformation MgCu2→ TbFe2

Equal positions for only part IIIa Multiple substitution CaCu5→ ThMn12, Zr4Al3

of the atoms IIIb Redistribution of the atoms Th2Ni17→ Th2Zn17

or substitution of an atomic group

IIIc Inclusion or elimination Cu → 𝛾’-Fe4N1, Mg → CdI2a, FeS2a

and redistribution of included atoms

All structure details (fragments) IVa Homeotectics (stackings CdMg3→ BaPb3, MgZn2→ MgCu2

are equal of closely packed networks)c

IVb Certain modes of stacking of slabs AlB2→ 𝛼-ThSi2a, Zr5Si4a→ Sm5Ge4a

(polyhedra), homogeneous homologous seriesd

Details (fragments) are only V Inhomogeneous homologous series,

partially equal interchanges different in two or more details

Va One-dimensional (AuCu3, CaF2) → HoCoGa5, Ho2CoGa8

Vb Two-dimensional Zr4Al3→ 𝜎 phase, P phase, 𝜇 phase

Vc Three-dimensional (AlB2, 𝛼-Fe) → Ce24Co11a

a _{These structure types do not belong to the family of close packing. They are shown for illustration of the transitions IIIc, IVb}

and Vc.

b _{External deformation is connected with changing of the 𝑐/𝑎 ratio.}

c _{The term homeotectic was suggested by Laves and Witte and is related to structures that have similar composition and equal}

CN but different ways of mutual placement of the fragments

d_{Homology suggests the possibility of the construction of structures with similar sets of fragments but with different quantitative}

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2.2.Phase stability ..

2

11

2.2. Phase stability

T

he stability of the crystal structures mentioned in the previous section will now bedescribed using thermodynamic properties. Especially for elastic or structural phase transitions commonly found in giant magnetocaloric materials, analysis of the Gibbs free energy has been proven to be most useful. The Gibbs free energy is given by

𝐺 = 𝐻 − 𝑇 𝑆 (2.1)

where 𝐻 is the enthalpy and 𝑆 is the entropy.

2.2.1. e role of entropy

The entropy can be expressed as

𝑆 = 𝑘𝐵𝑙𝑛𝑊 (2.2)

where 𝑘𝐵is the Boltzmann constant. Using this formulation, the entropy is dependent

on the number of possible conﬁgurations 𝑊 . In a highly symmetric system, a given atom can be distributed between a number (𝑔) of symmetry-related positions. Removal of a symmetry element then produces 𝑛 sets of 𝑔/𝑛 equivalent positions, reducing or eliminating the choice, and causing a decrease in 𝑆. This is beautifully illustrated for WO3a. WO3undergoes ﬁve transitions from low to high temperature: monoclinic →

triclinic → monoclinic → orthorhombic → tetragonal → hexagonal6_{. Apart from the} ﬁrst transition, this is exactly what is expected. Inchapter 4, Mn3Ga is discussed that

shows two successive phase transitions as a function of temperature.

2.2.2. e role of enthalpy: the Miedema model

The enthalpy of formation was modelled by Miedema et al.7_{to predict phase stability} of intermetallic compounds using a semi-empirical model. To evaluate the formation enthalpy Δ𝐻, two contact interactions at the interface of bulk metals are considered. The general assumption of this model is that these interactions dominate at the atomic level. When two metals are brought into contact with each other, the difference in work function will cause a charge to ﬂow, resulting in a dipole layer. This dipolar interaction corresponds to a negative contribution to the enthalpy of formation and is given for atom A in a matrix of B Δ𝐻𝑑𝑖𝑝= −𝑃 𝑉 2/3 𝐴 (ΔΦ)2/ < 𝑛 1/3 𝑊𝑆>, (2.3)

where 𝑉𝐴2/3is the contact-surface area, < 𝑛 1/3

𝑊𝑆 > is the average electron density at the

boundary of the Wigner-Seitz cell, ΔΦ is the difference in work function of A and B and 𝑃 is a constant. The second contribution to the interface energy is based on an electron density mismatch. In order to remove any discontinuities in the electron density, energy is needed to excite electrons to higher energy states. This positive term to the enthalpy of formation is given by Δ𝐻𝑠𝑜𝑙 = 𝑄𝑉 2/3 𝐴 (Δ𝑛 1/3 𝑊𝑆)2 (2.4)

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2

where 𝑄 is a constant. The interfacial enthalpy of A surrounded by B yields

Δ𝐻𝐴𝐵=

𝑉_𝐴2/3

< 𝑛1/3_𝑊𝑆>(−𝑃 (Δ𝜙

∗₎2_{+ 𝑄(Δ𝑛}1/3

𝑊𝑆)2), (2.5)

where 𝑃 and 𝑄 are dependent on the atomic species. The total entropy change can be determined by taking the concentration 𝑐 into account:

Δ𝐻 = 𝑐𝐴𝑐𝐵(𝑓𝐵𝐴Δ𝐻𝐴𝐵+ 𝑓𝐴𝐵Δ𝐻𝐵𝐴) (2.6)

where 𝑓𝐵𝐴represents the degree to which A is surrounded by B, given by

𝑓𝐴

𝐵 = 𝑐𝐵𝑠(1 + 𝛾(𝑐𝐴𝑠𝑐𝐵𝑠)2). (2.7)

The factor 𝛾 takes the values of 8, 5 and 0 for intermetallics, metallic glasses and solid solutions respectively and 𝑐𝑠𝐴is the surface concentration of A, given by

𝑐𝑠 𝐵= 𝑐𝐵𝑉 2/3 𝐵 𝑐𝐴𝑉 2/3 𝐴 + 𝑐𝐵𝑉 2/3 𝐵 . (2.8)

For the formation of stable alloys, Δ𝐻 < 0 and rewritingEquation 2.5leads to

ΔΦ ∝ 𝑄/𝑃 Δ𝑛𝑊𝑆− Δ𝐻 (2.9)

If ΔΦ is plotted versus Δ𝑛𝑊𝑆, like shown inchapter 7, one can separate combinations

of elements that prefer to mix and ones that do not form stable alloys. All values have been tabulated for elements in the metallic state7_{and can be found in the appendix.}

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2.3.Phase transitions ..

2

13

2.3. Phase transitions

A

s a single composition can exhibit various symmetries, we can now focus on the transitions between them. In general, phase transition processes can be divided into two classes: ﬁrst-order phase transitions and second-order phase transitions. First-order phase transitions show a discontinuous change in the order parameter and second-order phase transitions show a continuous change (the second derivative being discontinuous). A phase can be described by the free energy, that is dependent on thermodynamical variables such as temperature, pressure and magnetic ﬁeld. To do this, the potential is considered as a function of the crystal structure and can be characterized by the density function 𝜌(𝑥, 𝑦, 𝑧)8,9_{. The parameter of interest (order parameter) is therefore the} elec-tron density in the crystal. To describe the density in terms of the point group 𝐺0 at

the transition, it can be written down in terms of basis functions Ψ(𝑛)𝑖 of the irreducible

representations 𝜌 = ∑ 𝑛 ∑ 𝑖 𝑐_𝑖(𝑛)Ψ(𝑛)_𝑖 (2.10)

where 𝑛 is the number of the irreducible representation and 𝑖 is the number of the function at its basis. To describe any changes as a function of the crystal structure, 𝜌 is rewritten by isolating the identity representation

𝜌 = 𝜌0+ Δ𝜌 = 𝜌0+ ∑ 𝑛

′_∑ 𝑖

𝑐(𝑛)_𝑖 Ψ(𝑛)_𝑖 . (2.11)

The prime means that the identity representation is excluded from the summation. At the transition, either Δ𝜌 changes or Δ𝜌 = 0, which requires all 𝑐(𝑛)𝑖 = 0. The latter

corresponds to a second-order transition, where all the 𝑐(𝑛)𝑖 reduce to zero

continu-ously and take inﬁnitely low values near the transition point. In this case, the crystal transforms from a high symmetric point group 𝐺0to a low symmetric point group 𝐺1.

Despite the fact that the symmetry changes discontinuously, the physical properties re-main constant at the transition temperature. An example can be found inchapter 4. If the 𝑐𝑖(𝑛)coefﬁcients change discontinuously, the transition is of ﬁrst order. In the case

where 𝐺1is not a subgroup of 𝐺0, it is impossible to continuously transform one into

the other and the transition has to be ﬁrst order. Point group relations are described in

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2

Figure 2.2: Subgroup relations where the number of atoms in the unit cell is constant (no supercell formation). The straight lines indicate one-dimensional representations, the curves indicate multi-dimensional represen-tations8_.

To discriminate between ﬁrst- and second-order phase transitions, the analysis can be taken further by considering the thermodynamic Gibbs free energy 𝐺(𝜂, 𝑃 , 𝐻, 𝑇 ). In the vicinity of the transition, the potential can be described in terms of a polynomial expansion as function of the general order parameter 𝜂:

𝐺 = 𝐺0+ 𝑎1𝜂 + 𝑎2𝜂2+ 𝑎3𝜂3+ 𝑎4𝜂4+ 𝑎5𝜂5+ 𝑎6𝜂6+ ... (2.12)

where 𝐺0is the equilibrium value of the thermodynamic potential at the critical

tem-perature. When considering a structural phase transition, 𝜂 corresponds to 𝑐(𝑛)𝑖 , in

mag-netic phase transitions 𝜂 corresponds to the magnetization 𝑀 . The evaluation of these so-called Landau coefﬁcients yields 𝑎𝑖= 0 for speciﬁc values of 𝑖8,9. This discussion will

prove useful in discriminating between ﬁrst- and second-order magnetic phase transi-tions. To describe a magnetic system, the change in free energy can be described by

Δ𝐺 = 𝛼 2𝑀 2₊𝛽 4𝑀 4₊𝛾 6𝑀 6_{− 𝜇} 0𝐻𝑀 . (2.13)

where 𝜇0𝐻 is the applied ﬁeld and 𝛾 > 0. At the transition, the potential can be

evalu-ated considering 𝜕𝐺/𝜕𝑀 = 0:

𝛼 + 𝛽𝑀2_{+ 𝛾𝑀}4_{= 𝜇}

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2.3.Phase transitions ..

2

15 The parameter 𝛼 = 𝛼0(𝑇 −𝑇0) with 𝛼0> 0 ensures that 𝑀 ≠ 0 for 𝑇 < 𝑇0and 𝑀 = 0

for 𝑇 ≥ 𝑇0. The other coefﬁcients are considered to be temperature independent. The

equation can be solved easily in the absence of a magnetic ﬁeld. The coefﬁcients can be determined experimentally using an Arrott plot10,11_{, where 𝜇}

0𝐻/𝑀 is plotted as a

function of 𝑀2_{. When 𝛽 < 0, there are three minima and the free energy diagram}

looks likeFigure 2.3. This corresponds to a ﬁrst-order transition. For 𝛽 > 0 only two minima are found, corresponding to a second-order transition. The behavior of materials showing various values for 𝛽 is investigated inchapter 6.

Figure 2.3: Free energy difference ∆Φ as a function of the normalized magnetization 𝑚 = 𝑀/𝑀𝑠where 𝑀𝑠

is the saturation magnetization. In (a), 𝛽 > 0 and describes a second order phase transition, in (b) 𝛽 < 0 that describes a ﬁrst-order phase transition. (1) High temperature, with only the paramagnetic phase being stable; (2) the ferromagnetic phase is metastable but still separated from the paramagnetic phase by a free-energy barrier; (3) the free-energy barrier is gone and only the ferromagnetic state is stable.

The presence of a nucleation barrier is illustrated for ferromagnetic materials show-ing a ﬁrst-order magnetic transition. At high temperatures, the material will be in the paramagnetic state. Below a critical temperature 𝑇𝐶, the ferromagnetic state will have a

lower energy, but will not be formed initially. The kinetic barrier will retard the nucle-ation of the ferromagnetic state. Once the barrier is overcome, the ferromagnetic state will be readily formed and the magnetization will experience a jump. Since the entropy is the derivative of the free energy with respect to the temperature, it will also show a jump.

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2

2.3.1. Magnetoelastic coupling in hexagonal systems

In magnetic material systems there is a coupling between magnetism and unit cell pa-rameters. In case of a volume change, this coupling gives rise to the magnetovolume effect, where there is a volume contraction when the system becomes paramagnetic. This magnetoelastic coupling can be characterized by the Landau theory of phase tran-sitions by adding elastic terms and a bilinear magnetoelastic coupling to the free energy

Δ𝐺 = 𝛼 2𝑀 2₊𝛽 4𝑀 4₊𝛾 6𝑀 6_{− 𝜇} 0𝐻𝑀 + 1 2∑_𝑖,𝑘𝐶𝑖𝑘𝑒𝑖𝑒𝑘+ ∑_𝑖 𝜉𝑖𝑒𝑖𝑀 (2.15) where 𝐶𝑖𝑗are elastic constants, 𝑒𝑖is the elastic strain and 𝜉𝑖is the magnetic coupling

constant. The strain and magnetoelastic coupling can be expanded for a hexagonal sys-tem12_{. The (symmetric) elasticity tensor has the following form:}

𝐶_𝑖𝑗= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐶₁₁ 𝐶₁₂ 𝐶₁₃ 0 0 0 𝐶₁₂ 𝐶₁₁ 𝐶₁₃ 0 0 0 𝐶13 𝐶13 𝐶33 0 0 0 0 0 0 𝐶44 0 0 0 0 0 0 𝐶44 0 0 0 0 0 0 𝐶66 =12(𝐶11− 𝐶12) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Because 𝑒1= 𝑒2, 𝑒4= 𝑒5= 𝑒6= 0, the change in Gibbs free energy becomes

Δ𝐺 = 2(𝐶11+ 𝐶12)𝑒21+ 4𝐶13𝑒1𝑒3+ 𝐶33𝑒23+ 2𝜉1𝑒1𝑀 + 𝜉3𝑒3𝑀 (2.16)

and can be evaluated at the phase transition 𝜕Δ𝐺 𝜕𝑀 = (𝛼 + 2𝜉1𝑒1+ 𝜉3𝑒3)𝑀 + 𝛽𝑀 3_{+ 𝛾𝑀}5_{− 𝜇} 0𝐻 = 0 (2.17) 𝜕Δ𝐺 𝜕𝑒1 = 4(𝐶₁₁+ 2𝐶₁₂)𝑒₁+ 4𝐶₁₃𝑒₃+ 2𝜉₁𝑀 = 0 (2.18) 𝜕Δ𝐺 𝜕𝑒₃ = 4𝐶13𝑒1+ 2𝐶33𝑒3+ 𝜉3𝑀 = 0. (2.19) This yields three coupled equations. The bilinear magnetoelastic coupling in the ﬁrst equation effectively adds two terms to 𝛼. Assuming 𝜇0𝐻 = 0, this equation has a trivial

solution (𝑀 = 0) and non-trivial solutions, given by:

𝛼 + 2𝜉1𝑒1+ 𝜉3𝑒3+ 𝛽𝑀2+ 𝛾𝑀4= 0. (2.20)

This changes 𝛼 = 𝛼0(𝑇 − 𝑇0) inEquation 2.14and shifts 𝑇𝐶 to higher temperatures.

Similarly, linear quadratic terms will affect 𝛽. The remaining coupled equations result in 𝑒1 𝑀 = 𝜉1𝐶33− 𝜉3− 𝐶13 2𝐶2 13− 𝐶33(𝐶11+ 𝐶12) (2.21) 𝑒3 𝑀 = 𝜉3(𝐶11+ 𝐶12) − 2𝜉1𝐶13 2𝐶2 13− 𝐶33(𝐶11+ 𝐶12) . (2.22)

By measuring 𝑒 as a function of 𝑀 , the magnetoelastic coupling constants can be exper-imentally determined, and can be found inchapter 6.

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2.4.Magnetic properties ..

2

17

2.4. Magnetic properties

I

n this section, two models describing the properties of magnetic materials are dis-cussed. They are applicable to itinerant systems, that are of particular interest for magnetocaloric applications. Itinerant magnetic systems are especially suited for this purpose, due to the high concentration of magnetic atoms resulting in a potentially high magnetization. Within the group of materials with itinerant electrons, two kinds of magnetism can be distinguished. The ﬁrst is characterized by a very weak band splitting where the spin-up and spin-down states are very close to the Fermi level13_{, resulting} in a low magnetization and low transition temperature. The second is characterized by a strong band splitting, resulting in a high magnetization and high transition tempera-tures.

2.4.1. Saturation magnetization

The saturation magnetization determines the maximum magnetic entropy change and therefore strongly inﬂuences the magnitude of the magnetocaloric effect. The origin of the saturation magnetization in itinerant magnets is of quantum mechanical nature and is intuitively difﬁcult to understand. One approach is to use the chemical concept of bonding14_{. This will be explained using Fe as an example.}

At room temperature, Fe assumes a body-centered cubic (bcc) arrangement. In this arrangement, a central Fe atom is surrounded by 8 nearest neighbors at 2.5 Å and 6 next nearest neighbors at 2.9 Å. The atomic electronic conﬁguration is [𝐴𝑟]3𝑑64𝑠2_{, so}

at the Γ point three ﬁlled 𝑡2𝑔orbitals and two empty 𝑒𝑔orbitals are present. In a band

picture the high symmetry lines in the Brillouin zone are plotted as a function of energy. By summing these states over all orientations the density of states is obtained. The 𝑡2𝑔

orbitals have a better overlap, giving rise to broader bands compared to the 𝑒𝑔orbitals.

The total of three peaks in the DOS is characteristic for a bcc metal.

Figure 2.4: Band structure, DOS and Fe-Fe COHP of bcc iron, calculated without spin polarization. The dashed line in the DOS curve corresponds to the projected DOS of the Fe 4𝑠 orbital and its integration14_.

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2

Like in molecular bonding, for each two orbitals that are mixed, a bonding and an anti-bonding orbital are formed. This is illustrated by the Crystal Orbital Hamilton Population (COHP), where the DOS is separated into bonding (+) and anti-bonding (-) bands shown inFigure 2.4. In the case of Fe, there is occupation of anti-bonding states, destabilizing the crystal structure. One way of stabilizing the material is to adopt another crystal structure, but this does not happen. Instead the electrons shift in energy, splitting into a spin-up and spin-down band. Inchapter 5, we will show an example where a change in bonding results in a change band splitting, resulting in a change of magnetic moment.

Figure 2.5: DOS and COHP of bcc iron, calculated with spin polarization14. Spin-up and spin-down are indicated by the red and blue (dotted) lines.

A spin polarized calculation yields the spin-up (red, solid line) and spin-down (blue, dotted line) occupation (Figure 2.5). The shape of both densities of states is more or less the same, but they are shifted with respect to each other. The Fe-Fe bonding is now maximized by removing electrons from the anti-bonding band.

2.4.2. Transition temperature

For weakly magnetic systems, the Stoner model is used to describe the transition tem-perature. For systems like ZrZn2, the transition temperature is determined by exciting

spin-up electrons into the spin-down band by increasing temperature. The 𝑇𝐶is given

by

𝑇2

𝐶 = 𝑇𝐹2(𝐼𝑠𝑁 (𝜖𝐹) − 1) (2.23)

where 𝑇𝐹 = 𝜖𝐹/𝑘𝐵is the Fermi degeneracy temperature corrected for the effective

electron mass and 𝐼𝑠is the Stoner exchange parameter averaged over the one-electron

states.

In strong magnetic systems the Stoner model breaks down and leads to an overes-timation of the Curie temperature. For these systems, the magnetic moments persist above the Curie temperature but lose long range order. Therefore, spin ﬂuctuations are used to describe such a system, replacing the magnetic moment by the spin density. Then, using a Landau expansion, the free energy can be modeled15_.

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2.4.Magnetic properties ..

2

19 Using this model, one can deﬁne a spin ﬂuctuation temperature 𝑇𝑆𝐹

𝑇_𝑆𝐹 = 𝑀

2 0

10𝑘𝐵𝜒0

(2.24) where 𝑀0 is the total magnetic moment of the unit cell in 𝜇𝐵 at 0 K and 𝜒0 is the

exchange enhanced susceptibility at equilibrium 𝜒−1 0 = ( 1 4𝜇2 𝑏 ) ( 1 𝑁+_(𝜖 𝐹) + 1 𝑁−_(𝜖 𝐹) ) − 2𝐼_𝑠 (2.25)

where 𝐼𝑠is the Stoner exchange parameter. For materials where the critical temperature

is dominated by spin ﬂuctuations, 𝑇𝐶 is approximated by 𝑇𝑆𝐹. At the other extreme,

there is the Stoner theory, taking into account single particle excitations only, a descrip-tion which may be valid for systems where 𝑇𝑆𝐹is large compared with 𝑇𝐶. This leads

to the Mohn-Wohlfarth model 𝑇2 𝐶 𝑇𝑆2 𝐶 + 𝑇𝐶 𝑇𝑆𝐹 − 1 = 0 (2.26)

The validity of this model was investigated by plotting 𝑇𝐶/𝑇𝑆𝐹as a function of 𝑇𝐶𝑆/𝑇𝑆𝐹

for various materials inFigure 2.6. For Fe, Co and Ni, which have 𝑇𝐶values of 770, 1115

and 354∘C (1043, 1388 and 627 K) and magnetic moments of 2.2, 1.7 and 0.6 𝜇𝐵

respec-tively, the calculated values agree within an error or 3-5%. For Fe, the high magnetic moment leads to a high 𝑇𝐶. For Co, the increased density of states contributes to the

high 𝑇𝐶. For Ni, the lower 𝑇𝐶can be attributed to the reduced magnetic moment. The

model has been applied inchapter 6.

Although this model is not state of the art, it can approximate the Curie temperature quite well as long as there is no phase transition affecting the magnetism of the mate-rial. Extending this model by using a Heisenberg model and Monte Carlo simulations does not signiﬁcantly improve the predictions17_{. In addition, methods that use a ﬁnite} temperature DFT approach are still far from usable18_{. In case of phase transitions, other} DFT methods that model the ferro- and paramagnetic phases individually have been used successfully19_.

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2

Figure 2.6: The solutions ofEquation 2.26(full curve) and the actual values for 12 different systems (open circles). The broken line is the limit for pure Stoner behavior whereas the asymptotic limit for pure ﬂuctuation behavior is given by the chain line. The theory accurately describes the Curie temperatures of Fe, Ni and Co16_.

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References ..

2

21

References

[1] L. Pauling. ‘The nature of the chemical bond. IV The energy of single bonds and the relative electronegativity of atoms.’ J. Am. Chem. Soc. volume 54, p. 3570–3582 (1932).

[2] H. Q. Yuan, J. Singleton, F. F. Balakirev et al. ‘Nearly isotropic superconductivity in (Ba,K)Fe2As2.’ Nature volume 457, pp. 565–568 (2009).

[3] O. Tegus, E. Brück, K. H. J. Buschow and F. R. de Boer. ‘Transition-metal-based magnetic refrigerants for room-temperature applications.’ Nature volume 415 (2002).

[4] W. Zhang, A. R. Oganov, A. F. Goncharov et al. ‘Unexpected stable stoichiometries of sodium chlorides.’ Science volume 342, pp. 1502–1505 (2013).

[5] J. H. Westbrook and R. L. Fleischer. Intermetallic compounds I. John Wiley and sons ltd. (2000).

[6] H. A. Wriedt. ‘The O-W (oxygen-tungsten) system.’ Bull. Alloy Phase Diagr. volume 10, pp. 368–384 (1989).

[7] A. R. Miedema, P. F. de Châtel and F. R. de Boer. ‘Cohesion in alloys — funda-mentals of a semi-empirical model.’ Physica B+C volume 100, pp. 1 – 28 (1980). [8] B. K. Vainshtein, V. M. Fridkin and V. L. Indenbom. Modern Crystallography II.

Springer-Verlag (2000).

[9] L. Landau. ‘On the theory of phase transitions.’ Zh. Eksp. Teor. Fiz. volume 7, pp. 19–32 (1937).

[10] A. Arrott. ‘Criterion for Ferromagnetism from Observations of Magnetic Isotherms.’ Phys. Rev. volume 108, pp. 1394–1396 (1957).

[11] W. I. Khan and D. Melville. ‘Landau theory of magnetic phase diagrams and ﬁrst order magnetization processes.’ J. Magn. Magn. Mater. volume 36, pp. 265–270 (1983).

[12] J. F. Nye. Physical properties of crystals. Oxford university press (1957).

[13] P. F. D. Chatel and F. R. D. Boer. ‘The theory of very weak itinerant ferromagnetism applied to Ni3Ga and Ni3Al.’ Physica volume 48, pp. 331 – 344 (1970).

[14] G. A. Landrum and R. Dronskowski. ‘The orbital origins of magnetism: from atoms to molecules to ferromagnetic alloys.’ Angew. Chem. Int. Ed. volume 39, pp. 1560– 1585 (2000).

[15] P. Mohn. Magnetism in the solid state. Springer-Verlag (2003).

[16] P. Mohn and E. P. Wohlfarth. ‘The Curie temperature of the ferromagnetic transi-tion metals and their compounds.’ J. Phys. F: Met. Phys. volume 17, pp. 2421–2430 (1987).

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22 References

[17] N. M. Rosengaard and B. Johansson. ‘Finite-temperature study of itinerant ferro-magnetism in Fe, Co, and Ni.’ Phys. Rev. B volume 55, pp. 14975–14986 (1997). [18] S. Pittalis, C. R. Proetto, A. Floris et al. ‘Exact Conditions in Finite-Temperature

Density-Functional Theory.’ Phys. Rev. Lett. volume 107, p. 163001 (2011). [19] P. Roy, E. Brück and R. A. de Groot. ‘Latent heat of the ﬁrst-order magnetic

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3

Experimental methods and

instruments

3.1. Introduction

T

his chapter describes the sample preparation methods and the used instruments.Samples were prepared using an arc-melting furnace1_{and a melt-spinning device}2 followed by an annealing step. The crystallographic phases were characterized by X-ray and neutron diffraction. Magnetic properties were measured using various magnetome-ters. X-ray absorption experiments were performed at a dedicated setup in the European synchrotron facility at Grenoble, France.

Section3.2is dedicated to a new setup of an arc melting furnace, constructed to replace and improve an older instrument. It was constructed to ensure high quality samples, by minimizing contamination by oxidation3_{. Not only does oxidation alter the material} properties, in some cases it can increase the melting temperature considerably. In that case, a high electrical power is needed to produce these samples, which promotes evap-oration. High-quality samples are important because some systems are very sensitive to impurities. For instance, the superconductivity4_{or exotic magnetic states}5_{are strongly} affected by impurities and defects6_{. Relevant elemental properties like melting points} and formation energies of oxides are listed inchapter 8.

3.2. Arc melting furnace

3.2.1. Main chamber

T

he furnace is composed of a main chamber, copper crucible and torch (The bottom plate of the main chamber is set on a hydraulic lift and has three connec-Figure 3.1). tions. The ﬁrst (14) is the crucible connection that has a KF25 stainless steel co-seal. An O-ring with centering ring was not used because it promotes leakage due to the water pressure in combination with the high vacuum. The second connection is an electrical feedthrough (18) which is directly connected to the crucible, minimizing any electrical

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3

24 3.Experimental methods and instruments

resistance. The third connection is the vacuum outlet for the argon ﬂow, and is located behind 18 in the ﬁgure.

Figure 3.1: Schematic drawing of the setup. 1. Ar inlet 2. Torch cooling 3. Power connection (-) 4. Height manipulator 5. Handle 6. Vacuum seal 7. Z-bellow 8. Bellow insert 9. XY bellow 10. Paque insulation 11. Wobble stick 12. Glass cylinder 13. Crucible 14. Crucible connection 15. Alumina cap 16. Tungsten electrode 17. Paque insulation 18. Power connection (+) 19. Crucible cooling

The main chamber is composed of a glass cylinder (12) that has been cut to precisely ﬁt in the metal seal, which incorporates an O-ring. The glass rests on a soft aluminum layer that allows a vacuum level of at least 10−6mbar. The metal seals on the top and

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3.2.Arc melting furnace ..

3

25 bottom are kept together by paque strips (17) which are placed along the edge. The top part of the arc furnace is stationary and has been electrically isolated by a paque layer (10). The bellow insert (8) prevents the torch from making contact with the upper part, that can cause stray currents to affect the turbopump and the gauges. A wobble stick (11) is used as a sample manipulator to easily turn the samples. The torch is controlled man-ually by two handles (5), one on each side. The torch is sealed at (6) by two O-rings, that are pressed into a notch and held by a metal ring that is screwed into the stainless steel piece. The torch is composed of three layers, separating the two water ﬂow directions and the argon gas.

3.2.2. Vacuum system

The vacuum is controlled by a membrane pump (nXDS6i from Edwards) and a turbo-pump (nEXT240D from Edwards) and is monitored by a pirani-type vacuum gauge (AGP100), a wide-range gauge (WRG-S) and a piezogauge (ASG2). The pirani gauge can measure the pre-vacuum from ambient pressure to 10−3 mbar, the wide-range gauge can measure from ambient down to 10−9 mbar and the piezogauge can measure down to 1 mbar and is used to control the argon pressure. Two needle valves (which have a sensitivity of 0.1 L/min) control the argon flow, that is sustained by a separate small laboratory pump. A separate pump was used to create an argon flow, thereby preventing any gases formed in the melting process to enter the membrane pump. In addition, the argon flow is filtered by steel wool, that is used because of the low cost. The system is controlled by a programmable logic controller (PLC, type HTB1C0DM9LP) that has an integrated touchscreen. The pumps, valves and gauges are directly connected to the PLC.

3.2.3. Electrical system

A DC power supply (IMS TIG 161) with a low voltage (< 20 V) and an adjustable current between 10 and 160 A is used as generator. The maximum current is around 100 A for the standard 2.4 mm electrode (Binzel, E3 tungsten alloy) but is higher for thicker electrodes. The voltage of the IMS TIG 161 generator is given by Δ𝑉 (𝑉 ) = 0.04𝐼(𝐴) + 10 and the power is given by 𝑃 = Δ𝑉 𝐼. A current of 90 A can be sustained for any period of time, but 100 A can only be sustained for 0.6 minutes and 160 A can be sustained for 0.2 minutes without the machine overheating. Good electrical conductivity between the generator, the torch and the crucible is very important for the stability of the arc and the heating of the sample. The torch should be attached to the (-) side and the crucible to the (+) side. This way the electrons will travel from the crucible to the torch and loose their energy in the sample, heating it up. The other way around the electrode will heat up and melt above 25 A. Switching the connections could be a solution if the power is too high and one needs to gently heat the sample. To ensure a stable arc, the connection between (18) and the crucible needs to have a large contact surface and a tight connection.

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3

3.2.4. Cooling system

The cooling system consists of four valves and one flow meter. The first two valves are external and have to be operated manually. The water system will always run when the external switches are turned on and this enables cooling of the turbopump. The other two valves are solenoid valves (type EV210B from Danfoss) and allow water into the crucible and torch. The flow meter is capable of measuring a range between 0.5 and 30 L/min. The connections are push-in fittings from Festo; they allow easy removing of the tubes. The water flow inside the crucible was simulated using the Ansys Academic R14.5 software and is shown inFigure 3.2. Details on operating the instrument can be found in the manual.

Figure 3.2: Simulation of the velocity of the water ﬂow shown in a cross section of the copper crucible. The water (3.5 105Pa) enters the crucible from the center, disperses at a notch on the inner wall and exits from the side. This design ensures an equal distribution of water across the inner surface.

3.3. Melt spinning

T

o study metastable phases, the melt spinning technique was used. Intermetallic com-pounds in the solid state exist either as a solid solution, where any composition is stable, or have a ﬁxed stoichiometry. In the liquid state at sufﬁciently high tempera-tures, solutions can be made with any composition, and by rapidly quenching the liquid to room temperature it is possible to retain the starting composition. This is shown in

Figure 3.3. Using this technique, synthesis of metastable phases is attainable and can

in-crease the compositional range of a particular phase. This is the basis for melt spinning, where quenching is achieved by spraying a liquid onto a rotating copper wheel, achiev-ing coolachiev-ing rates in the order of 104- 107K/s7. The wheel is rotating at a speed of 60 m/s. The resulting ribbons can be crystalline or amorphous, depending on the elements present.

There are two additional advantages to this technique. First of all, arc melting can always leave some inhomogeneities in the sample due to the nonuniform temperature when melting. The induction furnace used in melt spinning makes sure that the samples are homogeneous. Secondly, hard materials are difﬁcult to process and powderization of samples is not always trivial. The resulting ribbons require short annealing times and

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3.4.Annealing ..

3

27

Figure 3.3: Schematic view of the melt spinning technique. A heating coil liquiﬁes a sample, that is sprayed through a nozzle onto a rotating copper wheel, creating (amorphous) ribbons8_.

are perfectly suited for characterization by X-ray diffraction. This is due to the high density and the fact that no force has been applied to the sample in order to achieve powderization. The melt spinner is a commercial product instrument by Edmund Bühler GmbH.

3.4. Annealing

T

he samples obtained by arc melting or melt spinning were annealed in a tube furnace.To protect the samples from oxidation, they were ﬁrst sealed into quartz ampoules in an Ar atmosphere of 200 mbar. The melt spun ribbons used inchapter 4were annealed at 650∘C for 2 h and quenched to room temperature. The samples fromchapter 6were annealed at 900∘C for 2 days and quenched to room temperature.

3.5. Diﬀraction experiments

D

iffraction is scattering of radiation from a periodic lattice, giving rise to interference.The interference pattern, or diffraction pattern, can be used to determine the crystal lattice. Each diffraction peak is determined by the scattering by one or multiple lattice planes, given by indices (ℎ𝑘𝑙). In these planes, the electrons (for X-rays) or nuclei and magnetic moments (for neutrons) of atoms 𝑛 are the source of the scattering. The sum of all the waves scattered by these atoms is called the structure factor 𝐹9_:

𝐹_ℎ𝑘𝑙=

𝑛=𝑁

∑

𝑛=0

𝑓_𝑛𝑒𝑥𝑝(2𝜋𝑖(ℎ𝑥_𝑛+ 𝑘𝑦_𝑛+ 𝑙𝑧_𝑛)). (3.1)

When the scattering objects are not localized in one point, the sum in the structure factor can be replaced by an integral over the volume 𝑉 :

𝐹_ℎ𝑘𝑙= ∭ 𝑉 𝜌_𝑥𝑦𝑧𝑒𝑥𝑝(2𝜋𝑖(ℎ𝑥 𝑎+ 𝑘 𝑦 𝑏 + 𝑙 𝑧 𝑐)) 𝑑𝑎 𝑑𝑏 𝑑𝑐. (3.2)

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3

where 𝑥, 𝑦 and 𝑧 are the fractional coordinates of a unit cell with cell axes 𝑎, 𝑏 and 𝑐. Now the electron density can be written as a Fourier series

𝜌_𝑥𝑦𝑧= 1 𝑉 ∑_ℎ ∑_𝑘 ∑_𝑙 𝑒𝑥𝑝(2𝜋𝑖(ℎ 𝑥 𝑎 + 𝑘 𝑦 𝑏 + 𝑙 𝑧 𝑐)). (3.3)

The structure factors were obtained by Rietveld reﬁnement10 _{using the FullProf} soft-ware11_.

3.5.1. Electron density plots

By using the structure factors as coefficients in a Fourier synthesis (performed by VESTA12 using a .fos file as input), the electron density can be reconstructed. The resolution of the electron density is given by the number of structure factors used. Due to the phase problem, only the real part of the structure factor is measured, because the intensity of a reflection is given by 𝐼 = |𝐹2|. The imaginary part (due to the phase) is calculated using the crystallographic model and is determined by the atomic positions. The mag-nitude of the structure factor is partly determined by the atomic form factor 𝑓, which is a scalar derived from an isotropic electron density distribution. Because the electron density distribution in real solids is generally anisotropic, the atomic form factor should depend on ℎ𝑘𝑙. This is reflected in the fact that the observed and calculated intensities for different reflections always show some deviations. By using the observed intensities to describe the real part of the structure factor, the anisotropic nature of the atomic form factor (which does not contribute to the imaginary part) can be taken into account. To measure changes in electron density as a function of temperature, the electron density at two different temperatures can be reconstructed and the difference can be plotted. In this way, all common features are removed and the electron density difference can be visualized. This is complicated by the fact that thermal motion of atoms changes the diffracted intensity. High temperature increases the thermal diffuse scattering and decrease the intensity at the diffraction peaks. However, the total scattered intensity should remain constant (neglecting any absorption). This means that the sum of the structure factors over all lattice planes should be a constant. This constant was used as a scale factor in the electron density plots, considering that the integral over the electron density is equal to the number of electrons in the unit cell.

3.5.2. X-ray powder diﬀraction

Room temperature X-ray diffraction (XRD) measurements were performed on a PANalytical X’Pert PRO diffractometer. The temperature-dependent X-ray diffraction experiments were performed in 0.5 mm capillaries at the BM1A beamline at the ESRF using a wave-length of 0.68884 Å and a PILATUS2M area detector. Temperature control was achieved using a liquid nitrogen cryostream.

3.5.3. Neutron powder diﬀraction

Neutron diffraction measurements were performed on the new neutron powder diffrac-tometer PEARL of the TU Delft13_{. Data were collected at 78 and 405 K using the (533)} reﬂection of the germanium monochromator (𝜆 = 1.665Å). The sample was loaded under

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3.6.Magnetization measurements ..

3

29 argon in a 6 mm diameter air-tight vanadium container (0.15 mm thickness). Cooling was achieved using a stream of liquid nitrogen and a heat gun was used for heating. The sample was measured for 10 min per temperature step. The data treatment consisted of a correction for the detection efﬁciency.

3.6. Magnetization measurements

M

agnetization measurements were carried out in a superconducting quantum in-terference device (SQUID) magnetometer (Quantum Design MPMS 5XL). Sam-ples were put in a gelatin capsule, which was mounted in a transparent plastic straw. Temperature dependent measurements were performed with a sweep rate of 2 K/min. High-temperature magnetization measurements were performed in a VersaLab vibrat-ing sample magnetometer (VSM) with an oven function. This instrument can heat a sample from 350 to 1000 K. The sample was in contact with a resistive heater and ﬁxed by thermal paste.

3.7. X-ray Absorption Spectroscopy

A

part from scattering experiments, absorption experiments can also provide valuableinformation about the electronic structure of a material. By tuning the energy of the X-rays to an absorption edge, the attenuated beam can be measured14_{. In general,} the intensity is measured as a function of energy. X-ray absorption ﬁne structure spec-troscopy (XAFS) uses high-energy photons to excite electrons from a core level to an excited state. The energy of the various core levels (1𝑠, 2𝑠 etc.) is known and at a syn-chrotron facility the energy can be tuned to an absorption edge. Because core levels are not inﬂuenced by the chemical environment of the material, the excited state is of par-ticular interest. For compounds containing Fe, the interesting excitations are a dipolar coupling from 1𝑠 to 4𝑝 and a quadrupolar coupling between 1𝑠 and 4𝑑 orbitals.

The XAFS spectrum can be divided into two regions: XANES (X-ray Absorption Near Edge Structure) and EXAFS (Extended X-ray Absorption Fine Structure). The XANES region is determined by the absorption edge and depends on the ﬁnal energy state, which is highly inﬂuenced by the oxidation state. The EXAFS region is found at higher energies and here the excited electrons have additional kinetic energy. This causes oscillations due to scattering. By analyzing the oscillations, the local environment (electron density around the absorbing atom) can be probed. Because in the Fe2P structure Fe and Mn

preferentially occupy different lattice sites, the electron density around the two lattice sites can selectively be probed.

The experiments were performed on powders with particle sizes between 20 and 40 𝜇m that were mixed with boron nitride to control the attenuation and pressed into pellets. The measurements were performed in transmission mode. Boron nitride secures the structural properties of the material without inﬂuencing the measurement. A nitro-gen cryostream was used to cool and heat the samples with a temperature control within 2 K. For better statistics, two absorption spectra were recorded at each temperature.

All XAFS spectra were analyzed with the Athena and Artemis software packages15_. In reducing the data, the same set of parameters was utilized for each absorption edge. For the Fourier transform at the Fe edge, 𝑘 values between 2 and 13 Å−1 were used;

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..

3

30 References

for the Mn edge, 𝑘 values between 2 and 12 Å−1were used since higher 𝑘 values were inaccessible due to an overlap with the Fe-edge. In fitting the data, an Einstein model was used to calculate the static disorder. To simulate the XAFS data, 6 scattering paths were used per temperature, based on the experimental lattice parameters. This accounts for 96% of the partial occupations of crystallographic sites and occupational disorder. The coordinates of the atoms were taken from neutron diffraction data. The energy shift between the experimental data and the fit was determined per measurement and the disorder and amplitude were fitted for the whole temperature range. A total of 5 parameters were used out of 13 independent points per measurement for the first shell of Fe. For Mn there are 11 independent points for the first shell. All fits were performed in 𝑘 space.

References

[1] H. Davy. ‘The Bakerian Lecture: An Account of Some New Analytical Researches on the Nature of Certain Bodies, Particularly the Alkalies, Phosphorus, Sulphur, Carbonaceous Matter, and the Acids Hitherto Undecompounded; With Some Gen-eral Observations on Chemical Theory.’ Phil. Trans. Roy. Soc. volume A 97, p. 71 (1809).

[2] D. Pavuna. ‘Production of metallic glass ribbons by the chill-block melt-spinning technique in stabilized laboratory conditions.’ J. Mater. Sci. volume 16, pp. 2419– 2433 (1981).

[3] Z. Altounian, E. Batalla, J. O. Strom-Olsen and J. L. Walter. ‘The inﬂuence of oxygen and other impurities on the crystallization of NiZr2 and related metallic

glasses.’ J. Appl. Phys. volume 61, pp. 149–155 (1987).

[4] C. Pﬂeiderer. ‘Superconducting phases of f-electron compounds.’ Rev. Mod. Phys. volume 81, pp. 1551–1624 (2009).

[5] H. V. Lohneysen. ‘Fermi-liquid instabilities at magnetic quantum phase transi-tions.’ Rev. Mod. Phys. volume 79, pp. 1016–1069 (2007).

[6] K. A. Gschneidner Jr. ‘Metals, alloys and compounds high purities do make a dif-ference!’ J. Alloy Compd. volume 193, pp. 1–6 (1993).

[7] R. W. Cahn. Physical metallurgy. North Holland (1983).

[8] N. T. Trung. ‘First-order phase transitions and the giant magnetocaloric effect.’ Doctoral thesis (2010).

[9] C. Hammond. The basics of crystallography and diffraction. Oxford University Press (2009).

[10] H. M. Rietveld. ‘A proﬁle reﬁnement method for nuclear and magnetic structures.’ J. Appl. Cryst. volume 2, p. 65 (1969).

[11] J. Rodríguez-Carvajal. Satellite Meeting on Powder Diffraction of the XV IUCr Congress volume 127 (1990).

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References ..

3

31 [12] K. Momma and F. Izumi. ‘VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data.’ J. Appl. Crystallogr. volume 44, pp. 1272–1276 (2011).

[13] L. van Eĳck, L. D. Cussen, G. J. Sykora et al. ‘Design and performance of a novel neutron powder diffractometer: PEARL at TU Delft.’ J. Appl. Cryst. volume 49, pp. 1398–1401 (2016).

[14] G. Bunker. Introduction to XAFS. Cambridge University Press (2010).

[15] B. Ravel and M. Newville. ‘ATHENA, ARTEMIS, HEPHAESTUS: data analysis for X-ray absorption spectroscopy using IFEFFIT.’ J. Synchrotron. Rad. volume 12, pp. 537–541 (2005).

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4

Mg

₃

Cd-based materials

4.1. Introduction

C

ompounds based on Mn and Ga can form a large variety of phases. The high spin polarization and moderate transition temperatures qualify them as good candidates for magnetocaloric materials. The structural and magnetic properties of Mg3Cd-type

(Mn1−𝑥Fe𝑥)3−𝛿Ga have been studied.

The Mg3Cd-type crystal structure is composed of two layers, each containing three

magnetic atoms (Mn, Fe or Ni) and one non-magnetic atom (Ga, In, Ge or Sn) in a hexagonal unit cell1_{. The magnetic atoms are arranged in a triangle in each layer, with} a non-magnetic atom above/below the center of this triangle in the other layer, as shown

inFigure 4.1. This phase is commonly referred to as the 𝜖 phase. The hexagonal phase

is metastable and will form in a temperature range between a high-temperature cubic and a low-temperature tetragonal phase2_{, and can be stabilized by quenching.}

Figure 4.1: Unit cell of the Mg3Cd type crystal structure. Three magnetic atoms (M) are arranged in a triangle

in one layer and a non-magnetic atom (X) in the center of the triangle, in the alternate layer indicated by shading.

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..

4

34 4.Mg3Cd-based materials

The ﬁrst magnetic study on the 𝜖 phase in the Mn-Ga system showed a magnetic moment of about 3.0 𝜇𝐵for Mn3. The magnetic moments are arranged in a triangular

antiferromagnetic conﬁguration4_{. The triangular arrangement has been described using} an Ising model with the spins conﬁned to the basal plane5_{. It gives rise to a complex} antiferromagnetic structure that can be described by a combination of three magnetic sublattices, oriented at an angle of 120∘. This often leads to exotic properties such as spin ice, multicritical phenomena, and noncollinear ordering6_.

Most of the triangular spin systems investigated so far are insulators, while Mn3Ga is

an example of a metallic system. This makes this system very interesting. Additionally, there is a sudden change in magnetization that coincides with a distortion of the lattice at 𝑇𝑑. A hexagonal-to-orthorhombic reduction in symmetry has been reported to take

place at the distortion temperature 𝑇𝑑, but no evidence has been presented to conﬁrm

this7,8_{. Using high-resolution X-ray diffraction, we will show that the symmetry is} ac-tually reduced to monoclinic. Such a behavior is highly uncommon, Tb3Ag4Sn4is the

only other example of such a transition9_.

The outline of the remainder of the paper is as follows, ﬁrst, the phase stability of the 𝜖 phase is investigated in the presence of Fe and Si substitutions. To explore if the crystallographic transition persists, the magnetic properties of (Mn,Fe)3−𝛿Ga and

Mn3−𝛿(Ga,Si) are reported. Finally, the origin of the crystallographic distortion is

in-vestigated using both X-ray and neutron diffraction techniques.

4.2. Experimental

M

n chips (99,9%), Ga pieces (99,99%) and Fe granules (99,98%) were melted in thearc melting furnace described in section 3.2. An extra 2 wt.% Mn was added to ac-count for evaporation losses; the evaporation varied between 1.75 and 2.25 wt.%, based on the change in mass. The obtained buttons were turned three times and subsequently melt spun to facilitate the phase formation. Combined evaporation losses from melt spinning and annealing result in a maximum deviation of 0.2 wt.% from the nominal stoichiometry. The melt-spun ribbons were ductile, which is an indication that the rib-bons are amorphous.

To find the transition temperatures of the tetragonal, hexagonal and cubic phases, differ-ential thermal analysis (DTA) was performed on a Perkin–Elmer MAS-5800 instrument with a heating/cooling rate of 10∘C/min and a nitrogen flow rate of 50 ml/min. We ob-serve transitions around 550, 600 and 700∘C. These preliminary samples were sealed in quartz ampoules, filled with Ar, and quenched in water. It was found that below 600∘_C

the tetragonal phase is stable and above 700∘_{C the cubic phase is formed. Apart from}

these phases, no other phases were observed after annealing at 450, 650 and 750∘_{C for 8}

h. All subsequent samples were annealed under argon at 650∘C for 2 h and quenched in water.

The diffraction experiments and magnetization measurements are described in section 3.5 and 3.6 respectively.

Electron density studies on magnetic systems

Delft University of Technology

Electron density studies on magnetic systems

ELECTRON DENSITY STUDIES

ON MAGNETIC SYSTEMS

MAURITS BOEIJE

ELEC

TRON DENSIT

Y S

TUDIES ON MA

GNETIC S

YS

TEMS

MA

URIT

S BOEIJE

Electron density studies on magnetic systems

Proefschrift

Maurits Boeije

Contents

1

Introduction

1.1.

Motivation

R

1.2.

Magnetocaloric eﬀect

M

1

1

1.3.

esis outline

B

1

References

1

2

eoretical background

2.1.

Crystal chemistry

I

2

2.1.1.

Crystal structures of intermetallic compounds

2

2

2

2.2.

Phase stability

T

2.2.1.

e role of entropy

2.2.2.

e role of enthalpy: the Miedema model

2

2

2.3.

Phase transitions

A

2

2

2

2.3.1.

Magnetoelastic coupling in hexagonal systems

2

2.4.

Magnetic properties

I

2.4.1.

Saturation magnetization

2

2.4.2.

Transition temperature

2

2

2

References

2

3

Experimental methods and

₃